STRUCTURES OF EQUALITY- AN INTRODUCTION

Parts Equal Total Structures of Equality-Copyright, 2012 Julie Russo

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Common%Core%State%Standards%for%Mathematics2%Common%Addition%and%Subtraction%Situations%

Table%2%Glossary;%page%88!! Result!Unknown! Change!Unknown! Start!Unknown!Add!To! Two!bunnies!sat!on!the!

grass.!!Three!more!bunnies!hopped!there.!!How!many!bunnies!are!on!the!grass!now?!

Two!bunnies!were!sitting!on!the!grass.!!Some!more!bunnies!hopped!there.!!Then!there!were!five!bunnies.!!How!many!bunnies!hopped!over!the!two!first?!

Some!bunnies!were!sitting!on!the!grass.!!Three!more!bunnies!hopped!there.!!Then!there!were!five!bunnies.!!How!many!bunnies!were!on!the!grass!before?!

Take!From! Five!apples!were!on!the!table.!!I!ate!two!apples.!!How!many!apples!are!on!the!table!now?!

Five!apples!were!on!the!table.!!I!ate!some!apples.!!Then!there!were!three!apples.!!How!many!apples!did!I!eat?!

Some!apples!were!on!the!table.!!I!ate!two!apples.!!Then!there!were!three!apples.!!How!many!apples!were!on!the!table!before!

!! Total!Unknown! Addend!Unknown! Both!Addends!Unknown!Put!Together/!Take!Apart!

Three!red!apples!and!two!green!apples!are!on!the!table.!!How!many!apples!are!on!the!table!

Five!apples!are!on!the!table.!!Three!are!red!the!rest!are!green.!!How!many!apples!are!green?!

Grandma!has!five!flowers.!!How!many!can!she!put!I!her!red!vase!and!how!many!can!she!put!in!her!blue!vase?!

!! ! ! !! ! ! !

What is a Structure of Equality?

¨  It is a graphic organizer. ¨  It is a representation of equality. ¨  It has values and labels. ¨  It organizes a math story. ¨  It is the thing you do between “Read” and “Solve”. ¨  It is used in conjunction with a prior taught strategy. ¨  It has a direct correlation with correct answers. ¨  It supports the 8 Mathematical Practices

What Does it Look Like-

Parts Equal Total 10 cookies

7 cookies (lemon)

3 cookies (orange)

Why Structures of Equality?

¨  How to Solve It, George Polya 1940 ¨  Singapore Math “Learning from Singapore

Math” Leinwand and Ginsburg Educational Leadership November 2007

¨  Research Into Practice MATHEMATICS “Solving Word Problems Developing Students’ Quantitative Reasoning Skills” Randall Charles

¨  Foundations of Mathematics-Valerie Faulkner ¨  Knowing and Teaching Mathematics Liping Ma

Why Structures of Equality- Again

The better the students are at recognizing the problem structure and representing it, the better their ability to solve more complex math problems. (Stigler, Fuson, Han, Kim 1986)

Common Question…

So are you telling me that I should not teach children to use a variety of strategies for math problem solving?

Does It Work? Mid-Year Wake County

Benchmark Testing

Average Scores for Algebra Strand

2011-2012

WCPSS: 74.3 KES: 75.1 Chris: 91.4

Wake County Mathematics Benchmark Testing-Laura’s 3rd Grade Classroom at

Kingswood Elementary 2012-2013

Projected Percent Proficient

Benchmark Test 1- 34.8

Benchmark Test 2- 62.8

Benchmark Test 3- 93.3

Chris McClellan- EOG Data

The First 4 of the 8 Practices

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics.

Video Clip

¨  As you are watching the video clips be thinking about the 8 Mathematical Practices.

¨  Focus on the first 4 Practices. ¨  Look for evidence of use throughout the video. ¨  Jot notes down on your paper. ¨  Be prepared to discuss what you noticed with a

partner.

Parts Equal Total-Video Clip

Turn and Talk

¨  Did you notice the use of any of the 8 Mathematical Practices?

¨  Were there Practices that were used more frequently the others?

¨  Give a specific example of when a Practice was used.

Parts Equal Total

¨  We can use a Parts Equal Total Structure when a math story describes:

*Composing 2 or more parts to form a total.

*Decomposing a total into 2 or more parts.

Example #1

Sarah and her 3 sisters share some toys. They have 4 dolls, 2 stuffed animals and one toy train. How many toys do they have in all?

Sample Questions

¨  Who’s the main character in this story? ¨  What’s the setting? ¨  What’s the verb or “doing words”? ¨  What’s the problem? ¨  Who’s job is it to solve the problem? ¨  What’s the label, unit, the thing we are counting, or

the common attribute? ¨  How many different types of toys are there? ¨  How many “parts” bars will we need?

Sarah and her 3 sisters share some toys. They have 4 dolls, 2 stuffed

animals and one toy train. How many toys do they

have in all?

What Does it Look Like?

They have 7 toys.

Sarah and her 3 sisters share some toys. They have 4

dolls, 2 stuffed animals and one toy

train. How many toys do they have in

all?

Example #2

¨  Example: Ms. Felder has 15 balloons. Three are red. Five are blue. The rest are yellow. How many balloons are yellow?

15  

Balloons    

3  

Balloons  (red)  

 

5  

Balloons  (blue)  

 

?  

Balloons  (yellow)  

 

3  +  5  =  8            15-­‐8  =7 There are 7 yellow balloons.

Rubrics

Structures of Equality Rubric Use of Structure

Label _____ (1 point) Value _____ (1 point) Equality _____ (1 point)

Equation _____ (1 point) Answer _____ (1 point) (Value and Label)

Work Habits Answer Circled ____ Name and Date ____ Correct Capitalization ____ Correct Punctuation ____ Complete Thought ____

You Try!

The Johnson family planted 15 pear trees in their yard last year. They planted 13 apple trees in their yard this year. How many trees are in their yard?

Use of Structure

Label _____ (1 point)

Value _____ (1 point) Be sure to look over the rubric before you are “done”. Equality _____ (1 point)

Equation _____ (1 point)

Answer _____ (1 point)

(Value and Label)

Parts Equal Total Structures and the Math Textbook Kindergarten and First Grade

Math Expressions-Houghton Mifflin Harcourt

Parts Equal Total-You Try

Samantha had 15 marbles in her bag, 4 fell out. How many are left? Example 1st Grade Problem

Parts Equal Total Structures and the Math Textbook 2nd Grade

There are 8 bananas and some apples. There are 12 pieces of fruit in all. How many apples are there? Math Expressions-Houghton Mifflin Harcourt 2nd Grade unit 1 Lesson 1

Parts Equal Total Structures and the Math Textbook-3rd Grade

Math Expressions-Houghton Mifflin Harcourt

Structures of Equality and the Commutative Property

¨  http://www.educreations.com/lesson/view/equality/532474/?ref=link

Parts Equal Total Structures and the Math Textbook-4th Grade

Chan He is carrying a box that weighs 37 pounds. In the box there are 5 containers of equal weight, and a book that weighs 2 pounds. What is the weight of each container? Draw a Parts Equal Total Structure to model this story.

Math Expressions 4th grade Unit 1 lesson 7-Houghton Mifflin Harcourt

2 lbs

37 lbs =

35 lbs 2 lbs

7 lbs 7 lbs 7 lbs 7 lbs 7 lbs

Parts Equal Total and the Math Textbook 5th Grade

One angle measure in an isosceles triangle is 100 degrees. What is the measure of each of other angles?

Math Expressions-Houghton Mifflin Harcourt 5th Grade Unit 4 Lesson 2

180 o =

100o

100o 80o

40o 40o

Structures Reinforce Writing

A Second Grader Writes a 2 Step Problem

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Common%Core%State%Standards%for%Mathematics2%Common%Addition%and%Subtraction%Situations%

Table%2%Glossary;%page%88!! Result!Unknown! Change!Unknown! Start!Unknown!Add!To! Two!bunnies!sat!on!the!

grass.!!Three!more!bunnies!hopped!there.!!How!many!bunnies!are!on!the!grass!now?!

Two!bunnies!were!sitting!on!the!grass.!!Some!more!bunnies!hopped!there.!!Then!there!were!five!bunnies.!!How!many!bunnies!hopped!over!the!two!first?!

Some!bunnies!were!sitting!on!the!grass.!!Three!more!bunnies!hopped!there.!!Then!there!were!five!bunnies.!!How!many!bunnies!were!on!the!grass!before?!

Take!From! Five!apples!were!on!the!table.!!I!ate!two!apples.!!How!many!apples!are!on!the!table!now?!

Five!apples!were!on!the!table.!!I!ate!some!apples.!!Then!there!were!three!apples.!!How!many!apples!did!I!eat?!

Some!apples!were!on!the!table.!!I!ate!two!apples.!!Then!there!were!three!apples.!!How!many!apples!were!on!the!table!before!

!! Total!Unknown! Addend!Unknown! Both!Addends!Unknown!Put!Together/!Take!Apart!

Three!red!apples!and!two!green!apples!are!on!the!table.!!How!many!apples!are!on!the!table!

Five!apples!are!on!the!table.!!Three!are!red!the!rest!are!green.!!How!many!apples!are!green?!

Grandma!has!five!flowers.!!How!many!can!she!put!I!her!red!vase!and!how!many!can!she!put!in!her!blue!vase?!

!! ! ! !! ! ! !

Choose one of the common core situations and write a grade specific problem to match that story type.

Handouts

¨ SoE Trifold Brochure ¨ SoE Explained ¨ SoE Lesson Plan ¨ SoE Work Mat ¨ SoE Assessments

Follow Up Workshops Include

¨  Structures of Equality- Compare

¨  Structures of Equality- Repeated Equal Groups

¨  Structures of Equality- Writing

¨  Structures of Equality- Assessment

¨  Structures of Equality-

Putting it Altogether

Contact Information

Julie Russo julie@mckeehomesnc.com

Special Thanks To: ¨ Tanya Burns ¨ Chris McClellan cmcclellan@wcpss.net ¨ Laura Smith lsmith6@wcpss.net ¨ Valerie Faulkner vffaulkn@ncsu.edu

Copyright Permission

Julie Russo, math consultant and copyright owner, grants to educators the usage rights of the Structures of Equality Documents with the credit of “Copyright 2012 Julie Russo” to be included on all Structures of Equality documents. The term “Structures of Equality” and all items created to implement Structures of Equality are property of Julie Russo (*see note below). Julie Russo gives permission to duplicate her Structures of Equality handouts and ideas for educational purposes only. Others should not gain financial profit through the usage of her Structures of Equality handouts and ideas. *Tanya Burns is a co-creator of some Structures of Equality documents. She is considered part owner of the documents she helped create, therefore her name will accompany Julie Russo’s on said documents.

Structures of Equality-Copyright, 2012 Julie Russo

If you are interested in leaning more about Structures of Equality, please contact Julie at julie@mckeehomesnc.com.